Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqvrelsymb Structured version   Visualization version   GIF version

Theorem eqvrelsymb 34985
Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised and distinct variable conditions removed by Peter Mazsa, 2-Jun-2019.)
Hypothesis
Ref Expression
eqvrelsymb.1 (𝜑 → EqvRel 𝑅)
Assertion
Ref Expression
eqvrelsymb (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))

Proof of Theorem eqvrelsymb
StepHypRef Expression
1 eqvrelsymb.1 . . . 4 (𝜑 → EqvRel 𝑅)
21adantr 474 . . 3 ((𝜑𝐴𝑅𝐵) → EqvRel 𝑅)
3 simpr 479 . . 3 ((𝜑𝐴𝑅𝐵) → 𝐴𝑅𝐵)
42, 3eqvrelsym 34984 . 2 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
51adantr 474 . . 3 ((𝜑𝐵𝑅𝐴) → EqvRel 𝑅)
6 simpr 479 . . 3 ((𝜑𝐵𝑅𝐴) → 𝐵𝑅𝐴)
75, 6eqvrelsym 34984 . 2 ((𝜑𝐵𝑅𝐴) → 𝐴𝑅𝐵)
84, 7impbida 791 1 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   class class class wbr 4888   EqvRel weqvrel 34632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-refrel 34899  df-symrel 34927  df-trrel 34957  df-eqvrel 34967
This theorem is referenced by:  eqvrelth  34990
  Copyright terms: Public domain W3C validator